Classical orthogonal polynomials via a second-order linear differential operators

Abstract

Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, \(D: = \frac{d}{{dx}}\) and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre \(\{L_n^{(2)}\}_{n\geq0}\) and the Jacobi \(\{P_n^{(\alpha-2,2)}\}_{n\geq0}\)

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Acknowledgement

The authors would like to thank the referees for their corrections and many valuable suggestions. The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number No. (RGP-2019-5).

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Correspondence to Baghdadi Aloui or Wathek Chammam.

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Aloui, B., Chammam, W. Classical orthogonal polynomials via a second-order linear differential operators. Indian J Pure Appl Math 51, 689–703 (2020). https://doi.org/10.1007/s13226-020-0424-6

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Key words

  • Orthogonal polynomials
  • quasi-definite linear functionals
  • classical polynomials
  • differential operators
  • structure relations

2010 Mathematics Subject Classification

  • 33C45
  • 42C05